When the fight first starts, you're going to get knocked back and Alysrazor will run right down the center, cleaving as she goes. Get out of that, and run off to your appointed side. (Each tank should cover one half of the room; this is where your egg is dropped and where your worms will emerge.) Make sure to growl at the other tank if they attempt to cross into your territory. Consider also peeing in a corner somewhere to mark it as such.
While running out of the cleave, throw up Divine Guardian for that initial burst of raid damage. The hitbox on Alysrazor is huge, so get a few hits in on her. You should be able to safely do a full Holy Power cycle and get a Shield of the Righteous off.
Once Alysrazor goes up in the air, head over to the side of the opening she burst from that belongs to you, and DPS the druid that comes down a little. Build up 3 stacks of holy power and bank them for your Voracious Hatchling, whose egg should be landing at this point. Cast Inquisition to use up your holy power, then hit Divine Plea to charge up a Shield of the Righteous. Immediately begin melee swinging as soon as the hatchling becomes available.
Don't do any special attacks yet. Wait for that "Hatchling has imprinted on you" announcement -- 100% more damage is a go! -- and then let the Shield of the Righteous fly! Hopefully, it won't miss.
Maximizing your hatchling damage output
Let's take a step back and talk about how to up your damage done in this fight. For starts, make sure to follow the tips I laid out in my 4 ways to improve your threat without sacrificing survivability column.
Take note especially -- especially -- the "Push moar buttonz" advice. I can't tell you how critical not wasting GCDs here is. I shouldn't have to tell you, either! Minimize movement (don't run screaming all around the area that you have Gushing Wounds), always keep your front or side to the hatchling, and be careful which way you face it. The hatchling will cleave to apply Gushing Wounds, so you don't want to clip a healer with it.
In terms of itemization, if needed, consider upping your hit and expertise. Because the hatchling is level 87, you only need 24 expertise skill and 6% hit chance to cap those respective stats. You'll really be able to see the upper limit of what kind of damage you're capable of when RNG is severely minimized.
Make sure to pop wings right before you're going to hit Shield of the Righteous, to maximize how many of those shield slams you can get in. Theoretically, you should be able to get two wings per hatchling phase, so pop it early. Also, consider also making a macro that pops wings and chugs a Golemblood Potion. You'll only get one of those per encounter, though, so save it for when you're having a difficult time getting the hatchling to die.
Lastly, glyph Focused Shield. It's a huge damage gain.
Don't get cut down by your bird
As for your survivability when handling the hatchlings, you have two huge buffs/debuffs to watch for. One is the aforementioned Gushing Wounds, which is a debuff applied to you. The other is Tantrum, a buff on the hatchling. I made a power aura for both so I can let the healer know what's on my plate in terms of incoming damage and, likewise, to give me a heads up when I'm going to need to be extra careful about my damage intake.
For Gushing Wound, there isn't a ton you can do to mitigate it, but consider popping Holy Shield as it falls off and you're at the 50% level, so you won't dip too low as the healer races to top you off. Likewise, whenever Tantrum occurs, you want to blow a cooldown (Divine Protection if available, along with Holy Shield) to take some of the edge off. The worst is when your bird Tantrums right as Gushing Wounds falls off. Don't get caught with your pants down if that occurs.
As for when to let your hatchling feed on the baby Magmaws, I'm partial to spacing them out at set intervals. As the hatchling health drops to 80%, 60%, 40%, and then 20%, I feed them a worm. If you're not getting the hatchling down fast enough, the danger zones when the bird is Hungry should be safely spaced out.
Lastly, be super-careful not to be standing in the path of Alysrazor while she's dive-cleaving the raid. You do not want to get cut down in the middle of that.
The danger zone
Eventually the hatchling will die, and the tornado phase will begin. Once you've successfully zipped your way around the flaming funnels (you know the deal -- follow a tornado, then as another passes, you jump into that lane and follow it, etc.), head toward the middle as Alysrazor crashes to the ground.
You'll likely be assigned to pick up the Blazing Talon Clawshaper on your side. Hit him with Avenger's Shield as he beings to cast Ignition to silence him and make him run toward you. Guide him into the maelstrom of splash damage as the raid unloads on Alysrazor, and keep Rebuke and other interrupts handy to block casts as they come up.
When Alysrazor gets up, you or the other tank will want to pick her up. To make it a little easier to pick up aggro, consider focusing the Clawshaper and making a focus-interrupt macro, so you can DPS Alysrazor while keeping up with your interrupting duties.
Eventually Alysrazor will become "reborn in flame" and hover above the raid. Guide her to face away from everyone, as she will cleave with Blazing Claw. (She loves to cleave.) Each swipe also increases your Fire and Physical damage taken by 10%, so swap off with the other tank about halfway through. Don't take more than 10 stacks.
Rinse and repeat ... in flames
Alysrazor will then fly up into the air once her Molten Power stacks max out at 100. The fight then goes through the same old cycle; druids land, eggs drop, hatchlings spawn, you murder the only thing in this world that loves you. So on and so forth.
The key to this fight is endurance. You'll be running through the cycle at least twice, so you need to play it smart and hang in there for the long game. Don't make silly mistakes, apply enough damage to kill your hatchling without trading up any more survivability than absolutely needed, etc.
In the end, this is your fight. Enjoy it. Bathe in those big numbers. And when it's on serious farm, come back in two pieces of ret tier 12, two pieces of prot tier 12, and the Ruthless gloves, and do some stupid-gigantic DPS. You earned it, having the sucky jobs on all the other fights!
The Light and How to Swing It tries to help paladins cope with the dark times brought by Cataclysm. Check out our protection 101 guide and our suggestions for protection paladin addons.